A Commentary on the Passing Scene by
Robert Paul Wolff
rwolff@afroam.umass.edu

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Sunday, April 27, 2014

A WILD RANT ABOUT MARGINAL PRODUCTIVITY

In the wake of the Piketty tsunami, there have been a good
many references to marginal productivity, particularly to the claim that the
stratospheric salaries of the "supermanagers" are justified on the
grounds that they are merely earning their marginal product, which is therefore
[this is always a very large leap] fair.The low wages of common workers are explained by their low marginal
productivity.

I am now going to do a complicated dance on a high wire
without a net.If I go splat, I hope some of you more
knowledgeable than I will clean up the mess.I should explain that in the Spring of 1978, during one of my very rare
sabbatical semester leaves, I sat in on Donald Katzner's graduate
microeconomics course at the University of Massachusetts, plowing my way
through Henderson and Quandt and doing all the exercises [though not handing
them in to be corrected by the long-suffering Katzner, who put up graciously
with my presence.]My knowledge of the
subject of marginal productivity derives entirely from that thirty-six year old
experience, so you can see this is a real reach for me.I mean, I was out of my depth when I was
fifty-four and in my prime [as Miss Jean Brodie would say.]At this point, I am desperately twiddling my
toes to tread water and stay afloat.Ok,
enough excuses.

The concept of marginal productivity is somewhat
misleadingly simple, at least as it is explained to undergraduate students
taking an elementary Micro course.The
idea is supposed to be this:There are a
number of inputs into any production process -- land, raw materials, tools, and
of course labor.Clearly [this is so far
from being clear that it is actually almost always false, but never mind that
for a moment], if we hold all but one of the inputs constant, and add one more
unit of the remaining input, there will be some increase in output.That increase is called the output at the
margin, or the marginal output, and this marginal increase in output
attributable to the addition of one unit of a factor is called that factor's marginal product.We could as easily ask how much output would
be lost if we held all but one of the inputs constant and decreased the
remaining input by one unit.Intuitively,
this will not in general be the same quantity as the quantity of output gained
by the addition of an extra unit, but [and here we go again with the little
assumptions on which total ideological rationalizations are erected] if we
assume continuity of inputs and outputs and
continuity of the production function of the process [which is not the same thing at all as the assumption of continuity of
the inputs and outputs, but never mind for the moment], then as we make the
units smaller and smaller, the difference between the two vanishes.

Everybody still with me?

Now comes the thing that gives economists secret erotic
thrills and convinces them that they are not merely scientists but Philosopher
Kings fit to rule a modern society.An
eighteenth century Swiss chap named Leonhard Euler [a very big deal in Mathematics] proved a theorem about what are
called homogeneous functions.Briefly,
to make this as simple as possible, a homogenous function in a number of
variables is a function in which the sums of the powers to which the variables
are raised in each term are equal.Thus,
if we have a function in three variables, f(x,y,z)
= (x2yz3 + xy4z + x1/2y1/2z5)
then the function is homogenous of order 6, because if you add up all the
little superscripts or powers in each term (including the 1's, which I did not
bother to put in), in each term they total 6.

Brief pause to permit the math averse to get a breath of
fresh air or read a Dickinson poem.

Euler proved that if we take the partial derivatives of the
function, f, and multiply each one by
the value of that variable at a given point x = (x1 , y1,
z1), then the value of the function at that point times the order of
the homogeneity is equal to the sum of all those partial derivatives multiplied
by the value of the variable at that point.Or, in symbols, where m = the order of homogeneity:

mf(x1,
y1, z1) = (df/dx1)x1
+ (df/dy1)y1 + df/dz1)z1

Now, when the order m = 1, the function f is called a linear
homogeneous function, and in this case:

f(x1, y1,
z1) = (df/dx1)x1
+ (df/dy1)y1 + df/dz1)z1

Well, the term on the left is simply the output of the product
for some level of inputs of x, y, and z, and the each term on the right can be
interpreted as the marginal product
of that factor, because it is the amount of output yielded by holding
everything else constant and increasing or decreasing that input marginally.So the total output of the product for the
inputs x1, y1, z1 is just exactly equal to the
sum of their marginal products.Now, the
income or receipts from selling the product is of course the quantity sold
times the price.If we pay each input
into the production of that output an amount proportional to its marginal
product, then the entire monetary return from selling the product will just
exactly be used up paying each factor of production its marginal product.In the simplest case, there are two factors
of production, capital and labor, and assuming that the society-wide production
function relating inputs of capital and labor to total social output is linear homogeneous, then clearly the
fair, efficient, rational, and one might go so far as to say divinely ordained
thing to do is to pay capital its marginal product in the form of profit, and
pay labor its marginal product in the form of wages.Ta Da!

It would not be an exaggeration, I think, to say, that when
economists looking for some way to justify low wages hit on this interpretation
of Euler's Theorem, it was as though the heavens had opened up and the heavenly
choir had started to sing to them.Pythagoras could not have asked for more.Pure Mathematics justified, indeed required,
that capital get its profits and labor be satisfied with its wages.

Sigh.It really is a
thing of beauty, this little bit of formal by-play.And thanks to the fact that most Americans
are essentially illiterate when it comes to math, it hasn't been difficult for
economists, who do really know better, to spend three quarters of a century or
so teaching their idiot students in Intro Econ courses that minimum wage laws
and taxation of profits and such like will just disturb the divine harmony of
marginal productivity and produce unimaginable inefficiencies and
counter-productive consequences.The
grad students learn better, but they also learn not to talk too much about what
they have learned, except to other initiates with doctorates in Economics.

Here's the thing.It
does not take much additional math to prove some rather unsettling conclusions
about economies governed by linear homogeneous production functions in which, to
be sure, Euler's Theorem applies.One
thing that falls out of the math is that such economies exhibit what is called
constant returns to scale, which means there is nothing to be gained by
expanding the scope of production in a company.That is equivalent to saying that the economy is in long run
equilibrium, which means no capitalist has any incentive to enter or exit any
particular line of production.It is
also true that in such an economy there is a zero rate of profit [not a zero
rate of interest -- that is something different].

Say what?Does this
sound at all like the American economy, or indeed any capitalist economy that
has ever existed?Well, er, no.But the math, the math, look at the math,
isn't the math beautiful?And it is
absolutely one hundred percent valid, that math.

It is all a fraud.A
beautiful fraud, an elegant fraud, but a fraud nonetheless.

Nobody in America is paid his or her marginal product, save
by accident, and we wouldn't know it if it happened, because there is no
plausible way of calculating the society's production function.What is more -- this is a topic for another
post -- the very notion of a continuous production function is about as close
to nonsense as you can get.[The great
Joan Robinson and her associates in Cambridge, England pointed this out long
ago.]

Well, if all of this doesn't drive you away from this blog
permanently, I will be very grateful.I
promise to return to more graceful and enjoyable topics tomorrow.

5 comments:

Your last equation f(x, y, z) = sum(diff(f(x,y,z), x..z)*(x..z)) is also the notorious trinity formula, if I am not mistaken. So, with Euler and the production function economics went back in time to Adam Smith.

Your own exposition is reminiscent of John Bates Clark (after whom the medal was named). In "The distribution of wealth: a theory of wages, interest, and profits" Clark conceived what for decades has been the standard Micro101 marginal productivity explanation: given a farm, if the farmhands work empty-handed (i.e. they don't use capital, or whatever capital used is negligible), we increase the number of farmhands and measure the output. The change in output, is the contention, is due to a change in labour only.

I have my own objection to that argument, but Joan Robinson and others came up with a simpler and humorous objection: if my production function uses shovels and muscular men to produce holes, the addition of a single shovel, without its corresponding strong man, will produce no additional output (i.e. hole). Similarly, with the addition of another man, without his own shovel.

Sure it does. Ask yourself whether an automobile plant could run without some sort of oversight and coordination. That is the function of management. That doiesn't mean it should be paid a thousand times as much as the work on the production line.

About Me

As I observed in one of my books, in politics I am an anarchist, in religion I am an atheist, and in economics I am a Marxist. I am also, rather more importantly, a husband, a father, a grandfather, and a violist.